Precision tube drawing for biomedical applications : Theoretical, Numerical and Experimental study

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Precision tube drawing for biomedical applications :

Theoretical, Numerical and Experimental study

Camille Linardon

To cite this version:

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THÈSE

Pour obtenir le grade de

DOCTEUR DE L’UNIVERSITÉ DE GRENOBLE

Spécialité : Matériaux, Mécanique, Génie Civil, Electrochimie

Arrêté ministériel : 7 août 2006

Présentée par

Camille LINARDON

Thèse dirigée parDenis FAVIER et Grégory CHAGNON

préparée au sein des laboratoires TIMC-IMAG (UMR CNRS 5525) et 3SR (UMR CNRS 5521)

et del’école doctorale IMEP2

Precision Tube Drawing for

Biomedi-cal Applications: TheoretiBiomedi-cal,

Numer-ical and Experimental Study

Thèse soutenue publiquement le , devant le jury composé de : M. Pierre-Yves MANACH

Professeur, Université de Bretagne-Sud, France, Président M. Frédéric BARLAT

Professeur, Pohang University of Science and Technology, Corée, Rapporteur M. Laurent DELANNAY

Professeur, Université catholique de Louvain, Belgique, Rapporteur M. Edgar RAUCH

Directeur de Recherche CNRS, Université de Grenoble, France, Examinateur M. Denis FAVIER

Professeur, Université de Grenoble, France, Directeur de thèse M. Grégory CHAGNON

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4.1.1.1 Geometry . . . 112 4.1.1.2 Material properties . . . 113 4.1.1.2.1 Tools materials . . . 113 4.1.1.2.2 Tube materials . . . 113 4.1.1.3 Boundary conditions . . . 114 4.1.1.4 Mesh . . . 114 4.1.1.5 Time incrementation . . . 114 4.1.1.6 Contact definition . . . 116 4.1.2 Isotropic models . . . 117 4.1.2.1 Mechanical model M1 . . . 117 4.1.2.2 Thermo-mechanical model M2 . . . 117 4.1.3 Anisotropic model . . . 118

4.1.4 Identification of the friction coefficient . . . 118

4.1.4.1 Identification of constant friction coefficients . . . 119

4.1.4.2 Analysis of the normal contact stress . . . 120

4.1.4.3 Identification of pressure dependent friction coefficients . . 120

4.1.5 Thermal contact properties . . . 122

4.1.5.1 Interfacial Heat distribution . . . 122

4.1.5.2 Thermal contact conductance . . . 124

4.1.6 Results of FEM . . . 124

4.1.6.1 Heat generation and exchanges . . . 124

4.1.6.2 Mechanical analysis . . . 128

4.1.6.2.1 General observations . . . 128

4.1.6.2.2 Influence of the die angle on the stress and strain fields . . . 129

4.1.6.2.3 Influence of the die angle on the drawing force and energies . . . 137

4.1.6.3 Influence of the die angle on the contact stresses . . . 139

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4.2.1 Homogeneous deformation method . . . 141

4.2.2 Slab method . . . 142

4.2.3 Upper bound method . . . 147

4.2.3.1 Energy dissipated by homogeneous deformation . . . 148

4.2.3.2 Energy dissipation due to internal shear . . . 148

4.2.3.3 Energy dissipation due to friction . . . 149

4.2.4 Drawing force . . . 152

4.3 Comparison of the analytical and FEM methods . . . 152

5 Failure prediction 157 5.1 Computation of the failure criteria from FEM . . . 158

5.1.1 Mechanical model considering isotropy, M1 . . . 158

5.1.1.1 Evaluation of failure criteria . . . 158

5.1.1.2 Influence of the die angle on Cockcroft-Latham failure criterion . . . 161

5.1.2 Mechanical model considering anisotropy . . . 163

5.1.3 Validation with 316LVM . . . 165

5.2 Discussion . . . 165

5.2.1 The different criteria . . . 165

5.2.2 My choice . . . 167

6 Conclusion and outlook 169

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General Introduction

Minitubes is a family founded company specialized in the manufacturing of precision tubes. The principal applications include biomedical devices such as surgical implants, stents and cardiac valves or a variety of in vitro diagnostic devices such as probes. Other applications in the field of aerospace or electronics for example exist but they are less challenging in term of precision compared to biomedical applications. Indeed, the components designed to be implanted in the human body require the tightest specifications.

From its foundation, Minitubes has developed a refined know-how in the field of tube drawing. Today, this know-how enables to reach the sharpest requirements and to satisfy clients demand. In the future, Minitubes intention is to formalise the process and to build a series of tools to define the different tube manufacturing process steps. More specifically, the intention is to optimise the process in order to increase the productivity and improve the product quality.

A better understanding of the process can be achieved by conducting large series of tests. Such approach happens to be time and money consuming due to the amount of raw material needed and more especially because the tests must be performed on the industrial drawing benches. At the industrial scale, experimental studies enable to easily measure different data, such as the drawing force, the temperature, the tube surface aspect and roughness, the final tube dimensions and the tube straightness. At a laboratory scale, due to the access to more complex analysis devices, experimental tests can give information about the material structure, texture, internal residual stresses, anisotropy and the heat generated due to plastic deformation to cite some of them. The combination of the possibilities offered by both environments can deliver rich information.

In this context, finite element modelling appears to be a helpful tool to improve the process understanding. The first interest of finite element modelling is to virtually perform a large number of tests. The second interest of numerical methods is that they give access to non measurable physical values such as strains and stresses during drawing. Such informations are necessary to improve the understanding of the process and above all to link the experimentally measured data to the internal phenomena taking place during material deformation.

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be performed in order to build it. First, as the material deforms during the process, its mechanical behaviour must be accurately characterised by means of laboratory tests. Any mechanical test can give stress vs strain data but it is fundamental to perform the laboratory tests in representative conditions compared to the industrial tube drawing. In the case where the variety of the testing devices is limited, it is crucial to evaluate the errors that can be made when simulating the process with simpler models. Second, like in any metal forming process, the material to be formed interacts with forming tools. Most of the time the contact is lubricated. This interaction phenomenon is important to be considered as it directly influences the drawing conditions and it can influence the deformations undergone by the material surfaces. Third, as the material plastically deforms and due to the friction between the material and the forming tools, heat is generated. If this phenomenon intends to be included in a finite element model, it must be characterised with care.

In the objective of process optimisation, a major point is to identify the material formability limit. In other words it signifies to determine the maximum deformation a tube can undergo before fracture. Once the experimental formability limit is known, the goal is to be able to predict it by means of finite element method. The challenge to predict tube failure is to select the appropriate tool among all the models and criteria that were defined by different authors. Due to the industrial requirements of selecting an efficient method and because of the limited mechanical testing techniques that were available at Minitubes, the choice was oriented towards failure criteria that could be calibrated on uniaxial tensile tests only.

The different topics dealt with in this thesis fit into the above described framework. The general objective of this study is to develop the finite element modelling of the tube drawing, first in order to improve the process understanding, second to find the formability limit and to optimise the process.

The first chapter is devoted to the presentation of all the notions involved in this study and to define the vocabulary. The principle of the tube drawing process is introduced and the different physical phenomena that are involved in such forming process are detailed. The different techniques that exist to analyse the drawing process are described and the focus is put on the interest of finite element modelling compared to analytical methods. Then, as one of the main concern is the material formability during tube drawing, the dif-ferent tools that were developed to study and to predict material formability are presented. This chapter ends with the description of a mechanical test called tube bulge test which is devoted to tube testing.

The second chapter presents the procedure that was used to characterise the materials mechanical and thermo-mechanical properties. The different testing techniques are intro-duced and the results are presented. The first objective of these experimental tests is to identify materials constitutive behaviour in order to model it. The second objective is to characterise materials failure.

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the purpose of this study. The originality of this test relies on the geometry of a drawing tool that was designed to draw tubes up to failure. The principle of this test is detailed and the different measurements performed during the test are described. From these tests, the material formability limit during tube drawing is identified.

The fourth chapter is devoted to the finite element modelling of the tube drawing process. First a general description of the model is made, second, the focus is put on the development of different models considering different aspects. Three models are developed: first, a purely mechanical one considering plastic isotropy, second, a thermo-mechanical one considering also plastic isotropy and last a pure mechanical model considering plastic anisotropy. The methods used to identify contact properties by inverse analysis are detailed. Finally, once the finite element model is fully defined, the tube drawing process is analysed in term of stress and strain fields and energies.

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Chapter

1 Introduction

Sommaire

1.1 Tube drawing process . . . 6

1.1.1 Introduction on tube drawing process . . . 6

1.1.2 Process parameters . . . 8

1.2 Phenomena to be modelled . . . 11

1.2.1 Plasticity . . . 11

1.2.2 Friction . . . 20

1.2.3 Heat generation and transfer . . . 25

1.3 Analysis of tube drawing . . . 29

1.3.1 Analytical methods . . . 29

1.3.2 Finite Element Modelling . . . 32

1.3.3 Comparison of the different methods . . . 34

1.3.4 Conclusion concerning the process analysis . . . 35

1.4 Formability . . . 35

1.4.1 Formability Limit Diagram . . . 37

1.4.2 Ductile fracture criterion . . . 38

1.5 Tube bulge test . . . 41

1.5.1 Strain measurement . . . 42

1.5.2 Stress computation . . . 45

1.5.3 Application of the bulge test . . . 45

1.6 Conclusions . . . 46

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the mechanical approaches that enable to analyse the process are detailed. Some analytical methods are briefly introduced and the insight is put into the Finite Element Modelling. A fourth part presents a major issue of metal forming industry, the metal formability. Indeed, the major concern of industry is to form parts safely which means without fracture occurrence. Finally, a test designed for the evaluation of mechanical properties of tubular materials is presented.

1.1 Tube drawing process

1.1.1 Introduction on tube drawing process

Cold tube drawing is a metalworking process used to produce high-quality seamless tubes with precise dimensions and good surface finish. Cold forming process compared to hot forming has three main advantages: tubes have more precise dimensions because it is not affected by thermal expansion, the surface finish is better and the mechanical properties are increased by strain hardening.

This introductory section first presents the different drawing techniques that are commonly used in the industry. Then the different operations required to manufacture the end product are explained.

1.1.1.1 Presentation of the different drawing processes

Cold tube drawing consists in reducing tube dimensions by pulling it through a die. There are four types of tube drawing process. For each of them, the tube outer diameter is calibrated by the die diameter. Their difference relies on the technique used for inner diameter calibration. The four kinds of tube drawing are tube sinking, mandrel drawing, floating plug drawing and fixed plug drawing. For illustration, the reader might refer to the figure 1.1 where the different kinds of tube drawing are shown. A brief explanation of each technique and their respective advantages and drawbacks are detailed below.

• Tube sinking consists in reducing the inner diameter with no tool inserted inside the tube. The inner surface is free to deform, as a consequence, the surface finish is degraded. The advantage of this technique is that it can be used in continuous drawing of coils.

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Figure 1.1: Four types of tube drawing (Yoshida and Furuya, 2004)

• Floating plug drawing is also known as floating mandrel drawing. It consists in inserting a specifically designed short plug inside the tube. The plug is free to move but stays located in the die vicinity due to friction forces between the mandrel and the tube. This process enables to reach a good surface roughness both inside and outside the tube. It can be used in continuous drawing of coils.

• In the fixed plug drawing the plug is fixed at the end of a rod. This technique is similar to the floating plug drawing and enables to reach the best surface finish.

1.1.1.2 Presentation of the drawing operations

Drawing a tube up to the wanted final dimensions requires several operations that are detailed here.

The very first tube to be drawn is manufactured by successive forging, rolling and drilling. This tube is called "ebauche". Starting from the ebauche to end up with the final product requires different successive drawing steps called passes. At each pass the tube is drawn to a certain section and thickness reduction. Between two passes, the tube is annealed to restore the material microstructure and ductility properties. The final passes are defined according to mechanical and metallurgical characteristics that are required by the client (ultimate tensile strength, yield strength, elongation, hardness, grain size). Finally, the process is ended by a straightening step to correct the curvature the tube has developed along the process. Figure 1.2 presents in a synthetic way an example of the operations required to manufacture a classic product in Minitubes

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12 - 10 - 6 - A 4 - 2 - 0 - 8 - Passes A A A A A A A A A A A A Radius ro ri A S

Mandrel drawing _Floating

plug drawing R R R R R R R R R _R R R reeling A annealing S straightening

Figure 1.2: Evolution of the tube dimensions throughout the drawing process and detail of the manufacturing operations. ro and ri are the tube outer and inner radii respectively. A

stands for Annealing step and S stands for Straightening step

The bibliography is very rich for the study of wire drawing but less developed for tube drawing. Nevertheless, both processes have common characteristics and some of the studies concerning wire drawing can be expanded to tube drawing. In the following section, various references concerning the wire drawing are cited but one has to keep in mind that the observations transfer to the tube drawing process.

1.1.2 Process parameters

The definition of a drawing pass requires to adjust different parameters. The section below enumerates the principal process parameters.

• The section and thickness reductions: ideally they should be the highest possible to limit the number of drawing passes. The first consequence of their increases is the increase of the drawing force. But the latter must not reach the bench limit capacity. Moreover the section and thickness reductions induce variations of plastic strain imposed to the material. Thus, the additional deformation necessary for the reeling step, which is compulsory after a mandrel drawing pass, may be greater. As a consequence, in this case, the risk of deteriorating tube dimensions and aspect during reeling is increased.

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Semi-cone angle, α

Bearing length

Die radius

Die entry radius

zoom

Figure 1.3: Die geometry

tensile strains (Sadok et al., 1994b) which do not contribute to the section reduction. It is the cause of the loss of the cylinder shape of the tube after drawing. An example illustrating the tube deformation is shown in figure 1.4. Redundant deformation appears as a deformation with an angle α positive at the tube extremity. The redundant deformation can be characterised by a redundant deformation factor φ which is the ratio of the average effective strain in the cross section of the material

avg on the homogeneous strain imposed in the drawing process h: φ = avg_h . The

factor φ depends only on the die semi-angle α and on the section reduction of the pass

RedS (Chin and Steif, 1995; Aguilar et al., 2002). Thus, it is common to compute a

parameter ∆ to combine both parameters. It can be expressed in different manners according to different authors but the numerical results differ little. As an example, Atkins and Caddel (1968) defined the ∆ parameter as:

1 +√1 − RedS

1 −√1 − RedSsin α (1.1)

And Backofen (1972) defined the ∆ parameter as:

∆ = α

RedS(1 −

√

1 − RedS)2 _(1.2)

Beland et al. (2011) analysed the influence of the die angle on the drawing force and revealed that an optimum die angle exists leading to a minimum drawing force. An example of the evolution of the experimental drawing force as a function of the die angle is shown in figure 1.5. But the drawing force must not be the only criterion to select a die angle. Indeed, die angle has a strong effect on the level of residual stress in the tube after drawing. Residual stresses are directly linked to the inhomogeneous deformation (redundant deformation). Depending on the application of the final product, residual stresses can influence the mechanical behaviour and the durability. For example, concerning wires, tensile residual stress at the wire surface can cause stress corrosion cracking and reduce the service time of the product (Elices et al., 2004; Överstam, 2006). Die angle is not the only responsible for the presence of residual stresses, one can also mention the heat generated during the process (Lee

et al., 2012). The phenomenon of heat generation is addressed in a further part.

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Ebauche Drawn tube

𝛼 > 0

(a) (b)

𝛼 > 0

Ebauche Drawn tube Drawing direction

Figure 1.4: Example of a tube before and after drawing showing redundant deformation

Figure 1.5: Influence of the die angle on the experimental drawing force during wire drawing (Beland et al., 2011)

performed tensile tests on wires drawn with the same reduction and different die angles and found that the yield and tensile strength increase with die angle.

• The drawing speed: it can influence the friction and the material behaviour if the material behaviour is viscoplastic.

• The lubrication: its role is to reduce friction between the tube and the drawing tools. It enables to prevent the occurrence of surface defects like scratches or wrenching. It is also a vector for heat extraction produced by plastic deformation and friction. The lubrication is dependent on the amount of lubricant, the nature of the contacting materials, their roughness, the sliding speed, the temperature and the pressure.

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1.2 Phenomena to be modelled

Tube drawing like any other metal forming process involves different phenomena that must be taken into account in a modelling. First, during forming, the material deforms in a irreversible way due to plastic deformation. Second, the material interacts with tools and the respective sliding of contacting materials causes friction. Finally, when a material plastically deforms and when there is friction between two materials, heat is generated. The generated heat then transfers to the contacting parts and to the surrounding environment. This part will be devoted to the description of the three phenomena to be modelled in metal forming process:

• plasticity; • friction;

• heat generation and thermal exchanges.

1.2.1 Plasticity

Metal forming is possible because of the material plasticity properties which is the ability of a material to undergo non-reversible deformations. The physical mechanism which is behind plastic deformation is the motion of dislocations. A dislocation is a linear defect corresponding to a discontinuity in the crystal organisation. The strain hardening is due to the accumulation of dislocations within the grains of a polycrystalline material. The dislocations can form different substructures depending on the nature of the material. Strain hardening is also due to the evolution of crystallographic texture. During plastic deformation, the tendency of the grains is to rotate towards more stable orientations and as a consequence, material hardening behaviour is modified.

Plasticity can be described by phenomenological models or by physically based ones. Phe-nomenological models are already implemented in FEM codes or can be easily implemented, and thus, they are convenient for industrial applications. Physical models are based on the theory of crystalline plasticity or on micromechanics. Models based on the crystal plasticity take into account the grain shape, the movement of dislocations within grains and the rotation of individual grains (Kalidindi and Schoenfeld, 2000; Van Houtte et al., 2002; Delannay et al., 2006; Li et al., 2008). All the physical phenomena at the origin of plastic deformation are modelled. Thus, the evolution of material anisotropy is naturally modelled.

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self consistent model in an UMAT subroutine. A UMAT is a subroutine that enables to model a user-defined mechanical material behaviour in Abaqus. The originality of their work is that they considered each integration point as a crystal with a given initial texture and followed texture evolution with deformation. But the main drawback of seft-consistent model combined with Hall-Petch only is that the strengthening due to dislocation density is neglected (Kapoor et al., 2010). Bui et al. (2013) developed a model to fill this gap. They modelled the strengthening due to both grain boundaries and substructures formed by dislocations and could predict the mechanical behaviour of cold drawn aluminium tubes up to various cross sectional reductions. The use of such models that enable to take into account the evolution of mechanical properties with deformation is important in the case of successive deformations analysis. For example, Karnezis and Farrugia (1998) analysed tube drawing by means of FEM and came to the conclusion that a two-pass tube drawing could be turned into a single pass. Bui et al. (2013) pointed out that Karnezis and Farrugia (1998) used the same phenomenological constitutive equation to study the two successive drawing passes and that, in this way, they did not take into account the change of mechanical properties of the tube after the first pass. They considered the mechanical properties of the tube being drawn at the second pass to be identical to the initial tube. As a comparison, Bui et al. (2011a) showed that a 36% section reduction of an aluminium tube caused the yield strength of the drawn tube to be three times higher than the initial tube and the elongation to be divided by four.

The main drawback of these models is that they require greater computational time and as a consequence they are less convenient for industrial applications.

Finally, phenomenological models are at the basis of this work due to their implementation into FEM codes and because of their ability to model material behaviour correctly. In a general way, phenomenological models are based on the definition of different consti-tutive equations that are detailed in the following section.

1.2.1.1 Plastic constitutive equations

Plasticity is commonly described by three constitutive equations which are a yield condition, a flow rule and a hardening law.

• The yield condition is described following a yield function which defines a surface in the stress space corresponding to the elastic limit and the transition to the plastic deformation. Its mathematical expression describes the shape of the yield surface. • The flow rule relates the stress and strain components and their time derivatives, it

gives the plastic strain rate.

• The hardening law describes the evolution of the yield surface during deformation in terms of expansion and translation.

Each of the above mentioned constitutive equations is detailed in this section.

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This material is said isotropic. On the contrary a material composed of directed grains exhibits properties that depend on the testing direction. This material is said anisotropic. Thus, constitutive equations can be classified into two categories depending on whether the material is isotropic or anisotropic.

1.2.1.1.1 Isotropic yield functions

The oldest isotropic yield function are the Tresca (1864) and the quadratic Von Mises (1913) yield criteria. Their respective expressions in the principal basis are the following:

Tresca : f = 1 2max(σi− σj) − σ0 2 (1.3) and Von Mises : f = 1₂q(σ1− σ2)2+ (σ2− σ3)2+ (σ3− σ1)2− σ0 (1.4)

σi and σj are principal stresses with i and j equals to (1, 2 ,3). Tresca expresses that

yielding occurs when the maximum shear stress reaches a constant critical value. σ0 is the yield stress in uniaxial tension.

Later Hosford (1972) extended the Von Mises yield criterion to a non-quadratic criterion based on polycrystal plasticity:

f = 1 2 |σ₂− σ₃|n_{+ |σ} 3− σ1|n+ |σ1− σ2|n) _n1 − σ₀ (1.5)

where n is a material parameter which is dependent upon the crystalline structure: for body-centered cubic (bcc) materials, n = 6 and for face-centered cubic (fcc) materials,

n= 8. Taking n = 2 returns the Von Mises expression.

1.2.1.1.2 Anisotropic yield functions

In some metal forming processes, materials are deformed in preferred directions. The consideration of material anisotropic behaviour is crucial to study any material forming process.

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Figure 1.6: scheme of the compression sampling and compression tests (Massé et al., 2011)

Direction of rolling

Figure 1.7: Example of grain elongation with reduction during sheet rolling (Park, 1999)

directions and anisotropy was induced. As a result the initial sheet showed anisotropic properties.

As a consequence, when forming materials, it is essential to know the whole deformation history in order to evaluate the material behaviour and to be able to model it with the appropriate constitutive equations. In order to explain the anisotropic behaviour, the material should be analysed at the microscopic scale.

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Some forming processes can produce complex textures such as gradient of texture in the thickness of the formed part. As an example, Park (1999) and Cho et al. (2006) showed that a gradient of texture was developing in the part thickness during cold sheet rolling and cold wire drawing respectively. Shear strain is said to be responsible for these variations. Indeed, Cho et al. (2006) found the shear strain to increase with the distance from the center line and found a gradient of texture between the center and the wire surface. Park (1999) noticed that friction at the sheet/roll interface during sheet rolling was causing shear deformation and as a consequence, the developed texture varied between the middle and the sheet surface.

As mentioned above, the forming process can induce anisotropic properties, thus, it is important to consider material anisotropy. The following paragraph lists anisotropic yield criteria proposed throughout the years.

The first anisotropic yield criterion was introduced by Hill (1948). Hill modified the Von Mises quadratic yield criterion by introducing coefficients to describe the plastic flow direction dependency. Hill’s expression is valid for orthogonal anisotropy and writes:

f = [F (σ22− σ33)2+G(σ33− σ11)2+H(σ11− σ22)2+2Lσ23+2Mσ31+2Nσ12] = σ02 (1.6) where σij are the components of the Cauchy stress tensor and F , G, H, L, M and N are

materials parameters. In the case of plane stress (σ33 = σ13 = σ23 = 0) the quadratic Hill yield criterion can be expressed as a function of the Lankford coefficients r0 and r90 which are the ratio of the width to the thickness strains ri = _thicknesswidth . The strains are

measured during tensile tests in 0◦ _{and 90}◦ _{with respect to the rolling or drawing direction} respectively. The yield criterion expression then turns:

f = σ2₁+r0(1 + r90) r90(1 + r0)

σ2₂− 2r0 1 + r0

σ1σ2 = σ20 (1.7)

where σ1 and σ2 are the principal stresses whose directions are aligned with the axis of anisotropy. σ1 is aligned with the rolling or drawing direction and σ2 is perpendicular. This criterion was extensively used in different studies and led to good results (Liao et al., 1997; Zang et al., 2011).

Afterwards, Hill generalized his own criterion (Hill, 1979) by introducing an anisotropy exponent m. Hill (1979) anisotropic yield criterion expresses in the space of principal stress:

f = [F |σ2− σ3|m+ G|σ3− σ1|m+ H|σ1− σ2|m+

L|2σ1− σ2− σ3|m+ M|2σ2− σ3− σ1|m+ N|2σ3− σ1− σ2|m] = σm0 (1.8) with σ1, σ2 and σ3 the principal stresses.

Hosford (1979) defined another yield criterion whose expression is similar to Hill (1979) and writes:

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is dependent on the anisotropic r values while it is independent in the case of Hosford (1979). Both Hill and Hosford non-quadratic anisotropic yield criteria are valid for planar/orthotropic anisotropy, when the directions of the principal stresses are superposed with the anisotropy axes. Their drawback is that they do not involve shear stresses. As a consequence, they cannot model the yield stress when the anisotropy axes do not coincide with the principal stress axes. Barlat and Lian (1989) introduced a new criterion to complete this limitation and defined a yield function which takes into account the shear stresses. Barlat and Lian (1989) yield function writes:

f = a|K1+ K2|m+ a|K1− K2|m+ c|2K2|m = 2σsm (1.10)

with K1 and K2 defined as:

K1 = σ11+ hσ22 2 and K2 = v u u u t σ11− hσ22 2 !2 + (pσ12)2 (1.11)

where a, c, h and p are the anisotropy coefficients and m is a non quadratic exponent depending on the material crystallographic structure as for Hosford (1979). In the special case of metal forming, 11 and 22 refer to the rolling or drawing directions and to the perpendicular to the rolling or drawing directions respectively.

As Barlat and Lian (1989) criterion includes the shear stress component, it can be used in cases were the anisotropy axes do not coincide with the stress axes. Nevertheless, this yield criterion is limited to plane stress problem. In order to solve three dimensional stress state problems, Barlat et al. (1991) proposed another yield criterion named Yld91 and extended the isotropic Hosford (1972) yield criterion to anisotropy by introducing a modified stress tensor ˜σ obtained from a linear transformation of the Cauchy stress tensor σ:

˜

σ = Mσ (1.12)

where M if a 4th-order tensor which due to the symmetry of the stress tensor can be reduced to a 6 × 6 matrix. In the case of an isotropic material, M reduces to the unit tensor. If the plastic behaviour is pressure independent, the stress deviator s can be used instead of the stress tensor. Similarly to the stress tensor, a modified stress deviator ˜s can be introduced by linear transformation of the stress deviator s:

˜s= Cs = CTσ = Lσ (1.13)

where C and L are fourth order tensors containing the anisotropy coefficients. T enables to transform the stress tensor σ into the deviatoric stress tensor s. In the case of orthotropic materials, the matrix of linear transformation writes:

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where a, b, c, f, g, h are six independent coefficients characterising anisotropy. The Yld91 yield criterion then writes:

f = |˜s2−˜s3|m+ |˜s3−˜s1|m+ |˜s1−˜s2|m = 2σm (1.15) Barlat et al. (1991) can be expressed in another form such as:

f =2 q H2 1 + H2 mh cos θ 3 −cosθ −2π 3 m + cos θ −2π 3 −cosθ+ 2π 3 m + cos θ+ 2π 3 −cosθ 3 m = 2σm _(1.16) with, θ= arccos q p3/2 ,0 6 θ 6 π (1.17) p= H₁2+ H2 (1.18) q= (2H₁3+ 3H1H2+ 2H3)/2 (1.19) and H1, H2 and H3 are the invariants of the transformed stress deviator:

H1= (˜s11+ ˜s22+ ˜s33)/3 (1.20)

H2= (˜s223+ ˜s231+ ˜s212−˜s22˜s33−˜s33˜s11−˜s11˜s22)/3 (1.21)

H3= (2˜s23˜s31˜s12+ ˜s11˜s22˜s33−˜s11˜s223−˜s22˜s233−˜s33˜s212)/2 (1.22) Barlat et al. (1991) revealed that this criterion could predict the uniaxial tensile yield stress in different directions but the accuracy was lower in the case of the Lankford coefficient prediction. Thus to improve predictions accuracy, Barlat et al. (2003) in 2000 proposed a new criterion called Yld2000-2d. This criterion is limited to plane stress state. Its expression is based on Hosford (1972)(1.5) isotropic criterion which is expressed as a function of the principal values of the stress deviator. Hosford (1972) yield function reduces to:

f = f0+ f00= 2σm where f0 = |s1− s2|m and f00= |2s2+ s1|m+ |2s1+ s2|m (1.23) The expressions of f0 _{and f}00 _{were transformed and expressed in terms of linear} transfor-mations of the stress deviator:

f0 = |˜s0₁−˜s0₂|m and f00= |2˜s₂00+ ˜s00₁|m+ |2˜s00₁+ ˜s00₂|m (1.24) where ˜s0

1, ˜s02 and ˜s001, ˜s002 are the principal values of the transformed stress deviators ˜s0 and ˜s00 respectively such as:

˜s0 = C0s= C0Tσ= L0σ (1.25)

˜s00= C00s= C00Tσ = L00σ (1.26)

Finally, the linearly transformed stress deviators can be written in a matrix form as:

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As a result, ten coefficients are necessary to describe plastic anisotropy. The procedure for parameters identification was given in Barlat et al. (2003). Nevertheless the Yld2000-2d criterion is limited to plane stress problems.

In order to solve three dimensions problems Barlat et al. (2005) proposed a new criterion named Yld2004-18p based on the combination of two linear transformations of the stress deviator. The expression of Yld2004-18p yield function is the following:

f = f(˜s0, ˜s00) = |˜s0₁−˜s₁00|m+ |˜s0₁−˜s00₂|m+ |˜s0₁−˜s00₃|m+ |˜s0₂−˜s00₁|m+ |˜s0₂−˜s00₂|m + |˜s0

2−˜s003|m+ |˜s03−˜s001|m+ |˜s03−˜s002|m+ |˜s03−˜s003|m (1.28) The different linear transformations applied to the stress deviators ˜s0 _{and ˜s}00 _are:

C0 =            0 −c0₁₂ −c0₁₃ 0 0 0 −c0₂₁ 0 −c0₂₃ 0 0 0 −c0 31 −c032 0 0 0 0 0 0 0 c0₄₄ 0 0 0 0 0 0 c0 55 0 0 0 0 0 0 c0 66            and C00₌            0 −c00₁₂ −c00₁₃ 0 0 0 −c00₂₁ 0 −c00₂₃ 0 0 0 −c00 31 −c0032 0 0 0 0 0 0 0 c00₄₄ 0 0 0 0 0 0 c00 55 0 0 0 0 0 0 c00 66            (1.29) In this case, the identification of the anisotropic coefficients requires a large number of experimental data such as uniaxial tensile test in seven directions between the rolling or drawing and the transverse directions and biaxial tests (Barlat et al., 2005). Yld2004-18p criterion was implemented by Yoon et al. (2006) in the FEM of cup drawing of a circular blank sheet and they successfully predicted the the cup heigh profile. They also showed the improved accuracy of Yld2004-18p compared to Yld96.

1.2.1.1.3 Flow rule

The flow rule gives the direction of the plastic strain rate and writes:

˙p ij = dλ

∂g

∂σij (1.30)

where ˙p

ij are the plastic strain rate components, dλ a scalar coefficient and g the dissipative

potential. This equation is called the non-associative flow rule as the dissipative potential

gis different from the yield function. In the case where the yield function f is taken as

the dissipative plastic potential, it is called the associated flow rule and writes:

˙p ij = dλ

∂f

∂σij (1.31)

1.2.1.1.4 Hardening constitutive equations

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Different material models can describe the work-hardening behaviour. The Hollomon’s equation writes:

¯σ = K¯n

p (1.32)

where ¯σ is the equivalent flow stress, K is the strength index, ¯p is the equivalent plastic

strain and n is the strain hardening exponent. Ludwik’s equation is generally preferred since it includes the yield stress σ0:

¯σ = σ0+ K¯np (1.33)

Voce law takes into account the variation of strain hardening exponent stating that the yield stress σ0 approaches a saturation value σs. The expression of the Voce law is the

following:

¯σ = σs−(σs− σ0) exp(−α¯)) (1.34) with α a dimensionless material parameter. Finally the Swift law can be used in the case of pre-strained materials as it is expressed as a function of a initial pre-strain ¯0.

¯σ = C(¯0+ ¯p)n (1.35)

The above expressions can be used in the case of both strain rate and temperature independent materials.

1.2.1.2 Viscoplastic constitutive equations

Other functions were developped to model visco-plastic materials behaviour. A comparative study of the different flow stress models was made by Banerjee (2007). In a general way, Johnson-Cook model is the most widely used. The Johnson-Cook model (Johnson and Cook, 1983) is an empirical relationship for the flow stress ¯σ which is described by:

¯σ = (A + B¯n p) 1 + C ln˙¯_p ˙¯0 1 − T∗m with T∗ ₌ T − T₀ Tm− T0 (1.36)

with ¯p the equivalent plastic strain, ˙¯p the plastic strain rate, ˙¯0 the reference plastic strain rate, A the yield stress, B the pre-exponential factor, n the work-hardening coefficient,

C the strain rate sensitivity factor, T is the temperature of the material, Tm the melting

temperature, T0 the reference temperature and m the thermal softening exponent.

1.2.1.3 Residual stresses

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Figure 1.8: Example of residual stress release by means of a destructive method (Kuboki

et al., 2008)

δ that are shown in figure 1.8. Figure 1.9 illustrates the process leading to the presence

of residual stresses and the deformation which is induced by their release. Figure 1.9(a) shows that inner and outer surfaces are successively loaded and unloaded up to different stress levels. The outer surface is unloaded and kept in an axial tensile state while the inner surface is unloaded and remained in an axial compressive state (fig.1.9(b)). The final state shown in figure 1.9(a) is taken as the initial state shown in figure 1.9(c) and the stresses are released. Outer surface stresses release leads to a negative axial strain while inner surface stresses release leads to a positive axial strain. As a consequence, tube outer surface shortens and the inner surface extends conducting to the deformed shape observed in figure 1.8.

Drawing methods (fixed plug drawing, mandrel drawing, tube sinking) lead to different levels of residual stresses (Yoshida and Furuya, 2004). Kuboki et al. (2008) showed that floating plug drawing compared to tube sinking could lower residual stresses. Photographies of the drawn tube can be seen in figure 1.8. In this figure, the tubes were cut to release the stresses, it is clear that the tube drawn with a floating plug deforms less which is the proof that the amount of residual stresses is lower in this case. Karnezis and Farrugia (1998) showed that turning a two-passes mandrel drawing process into a single-pass one to reach the same final tube dimensions could lower the residual stresses in the tube.

The residual stresses that can be present in tubes after drawing is a complex phenomenon that was not analysed in this study.

1.2.2 Friction

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A xi al st ress, σz Axial strain, εz Outer surface Inner surface A xi al st ress, σz Axial strain, εz Outer surface Inner surface Δεzo Δεzi Outer surface Inner surface Tensile stress Compressive stress Δεzo < 0 Δεzi > 0 Outer surface Inner surface (d) (b) (a) (c)

Figure 1.9: Explanation of the tube deformation due to the release of residual stresses

The normal load and the drawing speed are measured and controlled throughout the test. The conclusions of various studies are the following:

• the friction coefficient decreases with increasing sliding velocity (Nakamura et al., 1988; Kosanov et al., 2006; Szakaly and Lenard, 2010);

• the friction coefficient decreases with increasing normal contact pressure (Nakamura

et al., 1988; Emmens, 1997; Kosanov et al., 2006; Szakaly and Lenard, 2010). Ma et al.

(2010) developed a pressure dependent friction model based on the plastic deformation of surface asperities. Any surface which looks flat in appearance presents irregularities at the micro or nano scale: protrusions and depressions. When two irregular surfaces contact, and when the normal contact pressure increases, irregularities plastically deform and increase the friction surface which has a consequence on the apparent friction coefficient. For more information, the reader might refer to Szakaly and Lenard (2010) where the mechanism of lubrication is detailed;

• the friction coefficient increases with increasing material roughness (Emmens, 1997; Kosanov et al., 2006; Szakaly and Lenard, 2010);

• the friction coefficient decreases with the thickness of the lubricant film, i.e. the amount of lubricant (Nakamura et al., 1988; Emmens, 1997);

• the friction coefficient varies with the nature of the materials, harder tool materials induce lower friction coefficients (Szakaly and Lenard, 2010);

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• the friction coefficient depends on the temperature and the evolution of the friction coefficient with temperature can be described by a power law (Haddi et al., 2011):

µ µ0 = α _T T0 m (1.37)

where µ is the friction coefficient at the temperature T, µ0 is the friction coefficient at the reference temperature T0 and m and α are parameters.

The different studies that were listed above reveal the complexity of the frictional behaviour. The purpose of this study is not tribology but the concern is to identify a single friction coefficient value in order to use it as an input data into the tube drawing model. Thus, it is fundamental to identify a friction coefficient corresponding to the exact friction condition during tube drawing. Next section presents different experimental methods for friction characterisation.

1.2.2.1 Tests for friction coefficient characterisation

The experimental characterisation of the friction coefficient is complex. Some experimental tests were developed in order to evaluate the friction coefficient but their validity is uncertain as the tests are more or less representative of the experimental forming process. Lazzarotto et al. (1997) developed an experimental device to identify the friction coefficient in cold wire drawing. The as-developed upsetting sliding test consists in two parts extracted from real workpieces (a wire and an indenter) that are sliding relative to each other. The wire is fixed in a specimen stand linked to a tensile test machine. The indenter penetrates into the wire with a normal force Fnand slides on the wire with a sliding velocity v equal to

the drawing process one. During the test, the wire is plastically deformed by the indenter and the normal and tangential forces Fn and Ft are recorded. The friction coefficient is

expressed as a function of the radial spring-back of the wire behind the indenter δ, the penetration depth p, the contact length q and the measured normal and tangential forces. The expression is the following:

µ= δ − p+ qA

q −(δ − p)A with A = Ft

Fn (1.38)

Then Lazzarotto et al. (1997) introduced the as-determined friction coefficient in the Finite Element Modelling (FEM) of the wire drawing and found a relative error of 1% between the experimental and numerical drawing forces.

Vollertsen and Plancak (2002) present a push through test which is widely used for friction coefficient identification in the tube hydroforming process. A picture describing the test principle is presented in figure 1.11. In this method, a tube is expanded by an internal pressure against the tool. The tube is then pushed through the tool at a constant speed. As the tube slides inside the tool, a friction force is generated. It can be measured as a difference between the punch forces F1 and F2 or as a resulting force on the tool FR.

The friction coefficient is then obtained from the ratio of the measured friction force on the nominal contact force which is the contact area times the internal pressure.

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Figure 1.10: Design of the experimental sliding test and mechanical analysis (Lazzarotto

et al., 1997)

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Figure 1.12: Example of the tube upsetting test for friction coefficient identification (Vollertsen and Plancak, 2002)

presented in figure 1.12. It consists in upsetting a tube in a closed die while applying a internal pressure causing plastic deformation of the tube wall. The wall thickness then increases non-homogeneously due to the friction forces. The friction coefficient is then determined from the geometry of the tube wall with the help of analytical solutions or FEM.

The main purpose of the friction coefficient identification is to insert the identified values into a FEM. Next section introduces the main friction models that are currently used in contact modelling.

1.2.2.2 Friction model

There are two principal friction laws that are widely used to model the sliding behaviour between two contacting materials: the Coulomb and the Tresca models. To explain both models, it is convenient to define a contact between two rigid bodies A and B, sliding on each other at a velocity −→vs. Two stresses components act at the interface: a normal

contact stress σn and a shear contact stress σt which is tangent to the surface (fig.1.13).

The general expression of the Coulomb friction model is:

σt≤ µσn (1.39)

with µ the friction coefficient. The model defines a condition for sticking when σt< µσn

and a condition for sliding when σt= µσn. The general expression for the Tresca friction

model is:

σt≤ g (1.40)

with g a sliding threshold. g is a constant and expresses as g = m_√σ0

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A B 𝑣𝑠 𝜎𝑛 𝜎𝑡

Figure 1.13: Description of a frictional contact

1.2.2.3 Examples of friction coefficient

This section lists different examples of friction coefficient of a Coulomb model identified and used by different authors:

Reference Process studied Identification method µ

Majzoobi et al. (2008) Wire drawing FEM and analytical methods 0.035 to 0.15 Szakaly and Lenard (2010) - Experimental flat die tests 0.1 to 0.2 Karnezis and Farrugia (1998) Mandrel tube drawing Not mentioned 0.06 Yoshida and Furuya (2004) Floating plug drawing Not mentioned 0.1 Beland et al. (2011) Fixed plug drawing Not mentioned 0.035 Kuboki et al. (2008) Floating plug drawing Not mentioned 0.05

Table 1.1: Examples of used friction coefficient in different studies

Majzoobi et al. (2008) presented a range of friction coefficients depending on the nature of the lubricant. Szakaly and Lenard (2010) carried out a study to evaluate the effect of the normal contact pressure, the sliding speed, the nature of the contacting materials and the material roughness on the friction coefficient. In a general way, the range of friction coefficient used by different authors to study wire or tube drawing goes from 0.035 to 0.2 depending on the contacting materials, sliding speed, contact pressure and lubricant. Thus any study involving friction should be fed with a friction coefficient specifically identified to ensure its validity.

1.2.3 Heat generation and transfer

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1.2.3.1 Introduction

The basis of any thermal problem is the heat equation:

ρCp ∂T

∂t = ˙q + div(−−→qcond) (1.41)

where ρ is the mass density, Cp the specific heat capacity, ˙q a volumetric heat source

and −−→qcond the conduction heat flux vector. The right term of the above expression is the

combination of two components:

• The first component, ˙q, corresponds to a volumetric heat source defined as the thermal power per unit volume. In metal forming process, ˙q have two sources, plastic deformation and friction.

• The second component, −−→qcond represents the heat conducted from other domains that

are in contact with the domain of interest. The heat flux for conduction within a body is defined according to the Fourier law which states that the local heat flux vector −−→qcond (W m−2) for an isotropic material is equal to the product of thermal

conductivity k (W m−1_K−1_{) and the negative local temperature gradient (Km}−1_): −−→

qcond= −k

−−→

gradT (1.42)

1.2.3.2 Heat generated by plastic deformation

Mechanical energy used in cold working operations is converted both in heat and stored energy. The stored energy also known as the stored energy of cold work is in fact due to the creation or the rearrangement of crystal defects and the formation of dislocation structures.

The fraction of energy converted into heat is identified by the Taylor-Quinney coefficient β (Taylor and Quinney, 1933). β is equal to the ratio of the thermoplastic heating ˙Qp _{= β ˙}_Wp

on the plastic work rate ˙Wp= trace(σ ˙p). Thus ˙q in equation 1.41 can be replaced by:

˙q = β trace(σ ˙p₎ _(1.43)

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Material Test ˙, s−1 Max  β range βmean References

Steel (4340) Comp 3000 0.20 0.4 - 0.9 0.75 Mason et al. (1994) Steel (mild) Tors 0.0003 1.2 0.87 - 0.93 0.90 Taylor and Quinney (1933)

Steel (1018) Comp 3000 0.56 0.80 Kapoor and Nemat-Nasser (1998)

Aluminium (2024) Comp 3000 0.33 0.5 - 0.9 0.80 Mason et al. (1994)

Aluminium (3061) Comp 3000 0.72 0.85 Kapoor and Nemat-Nasser (1998) Table 1.2: Example of measured Taylor-Quinney coefficients (Comp and Tors stand for compression and torsion respectively)

More recently, Palengat (2009) computed an equivalent Taylor-Quinney coefficient by means of infrared measurements during tube tensile tests. Finally, Rusinek and Klepaczko (2009) estimated the fraction of plastic work converted into heat for TRansformation Induced Plasticity (TRIP) steels. This study is particular because added to the heat gener-ated by plastic deformation, there are also heat variations due to the phase transformation induced by plastic deformation.

1.2.3.3 Heat generated by friction

Two surfaces sliding on each other generate heat by friction. In most forming processes the heat which is generated is just a consequence of the process and is not specifically wanted. On the other hand, some processes use the heat generated by friction to assemble materials. It is the case for friction welding or friction stir welding were friction takes place at high speed. The power of friction Pf is defined as the product of the interfacial shear

stress τ and the sliding speed ||−→vs||:

Pf = τ||−→vs|| (1.44)

Only a part of this power is converted into heat and transmitted to the contacting materials. Heat is then distributed between the contacting surfaces according to a heat sharing coefficient f. The other part of the power is involved into wear phenomenon. Heat fluxes relative to a master qm and a slave qs surfaces write:

qm= fηPf and qs= (1 − f)ηPf (1.45)

with η the fraction of power of friction converted into heat.

The heat sharing coefficient is traditionally defined as a function of the material thermal effusivity ξ according to Vernotte (1956). The material effusivity expresses as function of the density ρ, the specific heat capacity Cp and the thermal conductivity k:

ξ =qρCpk (1.46)

Vernotte (1956) defines the heat sharing coefficient between two contacting materials as follow:

f = em

em+ es (1.47)

em and es correspond to the effusivies of the materials corresponding to the master and

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1.2.3.4 Surface thermal exchanges

Surface thermal exchanges can be divided into three distinct mechanisms: conduction, convection and radiation.

Concerning the interfacial heat conduction, a temperature drop is often observed at the interface between the two contacting surfaces. This phenomenon results from a thermal contact resistance existing between the surfaces in contact. Then, the thermal heat flux −→

qth between two contacting surfaces writes:

−→

qth = k(Tm− Ts)−→n (1.48)

where k is the contact thermal conductance, Tm and Ts the temperature of the master and

slave surfaces respectively and −→n the normal to the body surface.

Convection is the transfer of heat by the movement of fluids: the transfer takes place between a body and its environment. The heat flux by convection (−−→qconv) is described

according to the Newton cooling law which states that the heat loss of a body is proportional to the difference in temperatures between the body and its surroundings:

−−→

qconv= h(T − T∞)−→n (1.49)

with h the heat tranfer coefficient (W m−2_K−1_{), T the temperature of the body and T} ∞ the temperature of the environment, far from the body surface.

Radiation is the emission or absorption of electromagnetic radiation. The heat flux by radiation (−−→qrad) is described by the Stefan-Boltzmann law:

−−→

qrad= σ(T4− T∞4 )−→n (1.50)

with the material emissivity, σ the Stefan-Boltzmann constant equal to 5.67.10−8_{W m}−2_K−4_,

T the temperature of the body and T∞ the temperature of the environment.

1.2.3.5 Intermediate conclusion concerning thermal aspects

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1.3 Analysis of tube drawing

At Minitubes, the drawing passes are currently defined according to an empirical know-how. In a perspective of optimizing the production time, the concern of formalizing the process has grown. There are three techniques to conduct a process optimization study. First, tests can be conducted on small scale laboratory drawing equipments and the results can be transcribed at the industrial scale. Second, tests can be performed directly on the industrial drawing bench. Compared to the first method, it requires the interruption of the production and is more costly. Finally, the process can be modelled either by analytical methods of by FEM which is the main concern of this study.

As they are less time and money consuming, the last category will be presented in this section.

1.3.1 Analytical methods

Analytical methods are limited to the approximate expression of the drawing stress which is the ratio of the drawing force on the final tube section.

Deformation during tube drawing can be decomposed into three different components: the homogeneous deformation which depends only on the reduction ratio, an inhomogenous deformation also called redundant deformation linked to the geometrical parameters and finally the friction. Thus the work necessary for tube drawing can be written according to the work balance as:

W = Wh+ Wr+ Wf (1.51)

with Wh the work of homogeneous deformation, Wr the work of redundant deformation

and Wf the work of friction. Three analytical methods with respective advantages and

drawbacks are detailed and compared in this part: the homogeneous deformation method, the slab method and the upper bound method. These methods are presented in this order as they were developed with increasing complexity. Indeed, the homogeneous deformation method considers Wh only, the slab method includes Wf and the upper bound method

adds Wr

The presentation of the methods is limited to a brief introduction. The development of the methods and their application to tube drawing and in the specific case of this study will be detailed in chapter 4.

1.3.1.1 Homogeneous deformation method

This method relies on the hypothesis that all the work of external forces is converted into plastic deformation. An initial parallelepiped element transforms into a deformed parallelepiped element, no matter the intermediate deformations as shown in figure 1.14. In the first approximation the material is supposed perfectly plastic. The expression of the drawing stress is a function of the initial and final tube dimensions:

σd= F

Af = σ0ln Ai

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Initial parallelepiped element Final parallelepiped element A0 Af

Figure 1.14: Homogeneous deformation method applied to tube drawing (Rubio, 2006)

with σd the drawing stress, F the drawing force, σ0 the material yield stress and Ai and Af the initial and final tube sections respectively.

It can be seen that the drawing stress is expressed as a function of initial and final tube dimensions and as a consequence, it is independent of the die angle. It is a purely geometric method. The method can be more realistic considering material hardening by replacing the constant yield stress by a flow stress model. Finally, the homogenous deformation method is the simplest but it considers only the homogeneous deformation and neglects the friction and the redundant shear deformation. It idealises the process.

1.3.1.2 Slab method

The first development of the slab method was made by Siebel and Von Karman in 1924 and 1925 for the rolling process. Then Sachs (1927) was the first to investigate the slab method for the drawing process. The slab method is based on three principal assumptions: • the principal stresses do not vary on the planes perpendicular to the direction of the

applied load,

• frictional effects do not cause internal distortion of the material, • plane sections remain plane and the deformation is homogeneous.

In this method, a differential slab is considered within the deformed region. Figure 1.15 presents the different stresses acting on a slab element during mandrel drawing. The equilibrium of the stresses acting on the element is written considering both friction and homogeneous deformation. The drawing stress results from the integration of the as obtained expressions along the tube surface. The equilibrium equation of the slab in the z direction writes: σ1= (σ1+ dσ1)(t − dt) − σ2 dz cos α − µ1σ2dz − µ2σ2dz (1.53)

µ1 and µ2 are the die/tube and mandrel/tube friction coefficient respectively.

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z 𝛂

Figure 1.15: Stresses acting on an elemental slab during mandrel drawing (Kartik, 1995)

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(a) (b) (c)

Figure 1.16: Evolution of the predicted drawing force as function of the die angle: compar-ison of different analytical methods: (a) homogeneous deformation, (b) slab method, (c) upper bound method (Luis et al., 2005)

σ∗ is the uniaxial yield stress. More generally, the slab method can be seen as an

homogeneous method completed with friction.

1.3.1.3 Upper bound method

Luis et al. (2005) showed that both the homogeneous deformation and the slab method were unable to capture the effect of the die angle on the wire drawing force and the existence of an optimum die angle. Figure 1.16 shows the evolution of the predicted drawing force as a function of the drawing angle for the different methods they compared. The plot corresponding homogeneous deformation method shows no variation of the drawing force with the die angle and the slab method plot shows a decreased drawing force with increasing die angle, but no optimum appears. Figure 1.17 presents an example of chevron like fracture that can occur in wire during drawing or extrusion. Such a fracture is a direct observation of the axial stress heterogeneity in the wire and cannot be explained by the homogeneous and slab methods since these methods neglect shear. On the contrary, the upper bound method considers the shear introduced by the changes of direction of the material flow both at the die entrance and exit.

Precision tube drawing for biomedical applications : Theoretical, Numerical and Experimental study

HAL Id: tel-00956588

https://tel.archives-ouvertes.fr/tel-00956588

Precision tube drawing for biomedical applications :

Theoretical, Numerical and Experimental study

Camille Linardon

To cite this version:

THÈSE

DOCTEUR DE L’UNIVERSITÉ DE GRENOBLE

Camille LINARDON

Precision Tube Drawing for

Biomedi-cal Applications: TheoretiBiomedi-cal,

Numer-ical and Experimental Study

Contents

General Introduction

Chapter

1

Introduction

1.1

Tube drawing process

𝛼 > 0

𝛼 > 0

1.2

Phenomena to be modelled

1.3

Analysis of tube drawing

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